complexity results about nash equilibria vincent conitzer, tuomas sandholm international joint...
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Complexity Results about Nash Equilibria
Vincent Conitzer, Tuomas SandholmInternational Joint Conferences on Artificial Intelligence 2003 (IJCAI’03)
Presented by XU, JingFor COMP670O, Spring 2006, HKUST
2/18Complexity Results about Nash Equilibria (IJCAI’03)
Problems of interests
Noncooperative games Good Equilibria Good MechanismsMost existence questions are NP-hard for
general normal form games.Designing Algorithms depends on problem
structure.
3/18Complexity Results about Nash Equilibria (IJCAI’03)
Agenda
LiteratureA symmetric 2-player game and results o
n mixed-strategy NE in this gameComplexity results on pure-strategy Baye
s-Nash EquilibriaPure-strategy Nash Equilibria in stochasti
c (Markov) games
4/18Complexity Results about Nash Equilibria (IJCAI’03)
Literature
2-player zero-sum games can be solved using LP in polynomial time (R.D.Luce, H.Raiffa '57)
In 2-player general-sum normal form games, determining the existence of NE with certain properties is NP-hard (I.Gilboa, E.Zemel '89)
In repeated and sequential games (E. Ben-Porath '90, D. Koller & N. Megiddo '92, Michael Littman & Peter Stone'03, etc.) Best-responding Guaranteeing payoffs Finding an equilibrium
5/18Complexity Results about Nash Equilibria (IJCAI’03)
A Symmetric 2-player Game
Given a Boolean formula in conjunctive normal form, e.g. (x1Vx2)(-x1V-x2)
V={xi}, 's set of variables, let |V|=n
L={xi, -xi}, corresponding literals
C: 's clauses, e.g. x1Vx2, -x1V-x2
v: LV, i.e. v(xi)=v(-xi)= xi
G( ):=1=2= LVC{f}
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A Symmetric 2-player Game
u1(a,b) =u2(b,a)P2
P1L V C f
L1, li-lj
-2, li=-lj-2 -2 -2
V2, v(l)x
2-n, v(l)=x -2 -2 -2
C2, lc
2-n, lc -2 -2 -2
f 1 1 1 0
x1 -x1 x2 -x2
x1 1 -2 1 1
-x1 -2 1 1 1
x2 1 1 1 -2
-x2 1 1 -2 1
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Theorem 1
If (l1,l2,…,ln) satisfies and v(li) = xi, then There is a NE of G() where both players play li with pro
bability 1/n, with E(ui)=1. The only other Nash equilibrium is the one where both pl
ayers play f, with E(ui)=0.
Proof: If player 2 plays li with p2(li)=1/n, then player 1
Plays any of li, E(u1)=1
Plays –li, E(u1)=1-3/n<1
Plays v, E(u1)=1
Plays c, E(u1)≤1, since every clause c is satisfied.
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Theorem 1
No other NE: If player 2 always plays f, then player 1 plays f. If player 1 and 2 play an element of V or C, then a
t least one player had better strictly choose f. If player 2 plays within L{f}, then player 1 plays f. If player 2 plays within L and either p2(l)+p2(-l) <1/
n, then player 1 would play v(l), with E(u1)>2*(1-1/n)+(2-n)*(1/n)=1.
Both players can only play l or -l simultaneously with probability 1/n, which corresponds to an assignment of the variables.
If an assignment doesn’t satisfy , then no NE.
10/18Complexity Results about Nash Equilibria (IJCAI’03)
A Symmetric 2-player Game
u1(a,b) =u2(b,a)P2
P1L V C f
L1, li-lj
-2, li=-lj-2 -2 -2
V2, v(l)x
2-n, v(l)=x -2 -2 -2
C2, lc
2-n, lc -2 -2 -2
f 1 1 1 0
x1 -x1 x2 -x2
x1 1 -2 1 1
-x1 -2 1 1 1
x2 1 1 1 -2
-x2 1 1 -2 1
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Corollaries
Theorem1: Good NE is satisfiable.
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Corollaries
Hard to obtain summary info about a game’s NE, or to get a NE with certain properties.
Some results were first proven by I. Gilboa and E. Zemel ('89).
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Corollaries
A NE always exists, but counting them is hard, while searching them remains open.
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Bayesian Game
Set of types Θi , for agent i (iA)Known prior dist. over Θ1 Θ2…Θ|A|
Utility func. ui: Θi12…|A| RBayes-NE:
Mixed-strategy BNE always exists (D. Fudenberg, J. Tirole '91).
Constructing one BNE remains open.
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Complexity results
SET-COVER ProblemS={s1,s2,…, sn}S1, S2, …, Sm S, Si=SWhether exist Sc1, Sc2, … , Sck s.t. Sci=S ?
Reduction to a symmetric 2-player gameΘ= Θ1= Θ2={1, 2,…, k,} (k types each) is uniform= 1= 2={S1, S2, …, Sm, s1,s2,…, sn}Omit type in utility functions
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Complexity results
Theorem 2: Pure-Strategy-BNE is NP-hard, even in symmetric 2-player games where is uniform.
Proof:If there exist Sci, then
both player play Sci when
their type is i. (NE)If there is a pure-BNE,
No one plays si
{Si (for i)} covers S.
P2
P1Sj sj
Si 11, sjSi
2, sjSi
si
3, siSj
-3k,siSj
-3k